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MathGroup Archive 2004

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Re: LogIntegral^(-1)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg48682] Re: LogIntegral^(-1)
  • From: "Roger L. Bagula" <rlbtftn at netscape.net>
  • Date: Thu, 10 Jun 2004 02:43:39 -0400 (EDT)
  • References: <ca3gvi$rto$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

I really need to know if this is:
Li(x)^(-1)
or
ArcLi(x)
since it makes a great difference and Mathematica notation on this seems 
unclear.
One approximation ( Euler's I think) of the distribution of primes is
Pi(n)=Li(n)--> n/log(n): asymptotic
If it is "arc" that is very different than 1):

1) s[t_]=C[2]*(-1+1/Li[w*(t-C[1])/C[2]])

2) s[t_]=C[2]*(-1+ArcLi[w*(t-C[1])/C[2]])

1/Li(n)-->0 as n->Infinity
That is the signal dies out as time gets large.
But for Li[n]
ArcLi[x]=n
which becomes infinite as time goes on.
It is my conclusion that
it should be the first:
1/Li[n]--> 1-ArcTanh(n): approximation that tends the same way
But that gives:
s[t_]= -C[2]*ArcTanh[w*(t-C[1])/C[2]]
as the approximate signal function.
I don't like the implications of that.
Roger L. Bagula wrote:
> I have an information theory problem:
> Suppose that the noise has a constant expectation value with time
> so that when you take the differential you get
> DN/dt=0
> Mathematica for the Shannon capascity as rate of transmission is:
> s1[t_]=D[s[t],t]
> n[t_]=s[t]/(Exp[s1[t]/w]-1)
> n1[t_]=D[n[t],t]
> DSolve[FullSimplify[n1[t]]==0,s[t],t]
> 
> I get an answer involving:
> LogIntegral^(-1)
> specifically:
> 
> s[t_]=C[2]*(-1+LogIntegral^(-1)[w*(t-C[1])/C[2]])
> 
> I'm pretty sure that C[2] represents the average noise.
> The problem is I can't get Mathematica to plot this function.
> I replaced c2 and c1 with real constants and used a constant w
> while trying to plot Re[s[t]]:
> I get errors like:
> Plot::"plnr":
>      "\!\(Re[\(s[t]\)]\) is not a machine-size real number at \!\(t\) = \
> \!\(-0.999999916666666699`\)."
> Plot::"plnr":
>      \:f3b5\!\(Re[s[t]]\) is not a machine-size real number at t = \
> -0.918866016854168421`.
> Plot::"plnr":
>      "\!\(Re[\(s[t]\)]\) is not a machine-size real number at \!\(t\) = \
> \!\(-0.830382400281252586`\)."
> General::"stop":
>      "Further output of \!\(Plot :: \"plnr\"\) will be suppressed during 
> this \
> calculation."
> 


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